stats 2

Understanding Statistics and Data

  • Definition: Statistics is the study of data, including how to collect, analyze, and interpret information to make informed decisions.
  • Datasets: Composed of cases (subjects or units) which can be people, animals, or objects.
  • Variables: Characteristics of cases that can take on various values. Examples include:
    • Age
    • Gender
    • IQ Scores
    • Test Scores
  • Labels: Special types of variables used to uniquely identify cases (e.g., participant numbers), which hold no substantive meaning.

Measurement Levels

  • Categorical/Qualitative Variables:

    • Nominal: No order or measurable unit (e.g., ethnicity, gender).
    • Ordinal: Have an order but no measurable unit (e.g., family position: youngest, middle, oldest).

  • Quantitative Variables:

    • Interval: Ordered values with no fixed zero point (e.g., IQ).
    • Ratio: Ordered values with a meaningful zero point (e.g., age).

  • Precision of Measurement Levels: Nominal is the least precise; ratio is the most precise.

Data Analysis Process

  1. Identifying Variables:
    • Who?: Define the subjects or cases.
    • What?: Determine measurable characteristics (variables).
    • Why?: Understand the reasoning behind data collection.
  2. Data Distribution: After identifying variables, examine their distribution.

Displaying Data Distributions

Categorical Variable Distribution

  • Methods:
    • Pie Chart: Shows categories and their relative frequencies (should equal 100%).
    • Bar Graph: Displays frequency on the y-axis vs. variable values on the x-axis.

Quantitative Variable Distribution

  • Methods:
    • Histogram: Illustrates data distribution quickly using bars for intervals.
    • Stem Plot: Displays original data values divided into stems and leaves for clarity.

Describing Distributions

  • Measures of Center:

    • Mean: Sum of values divided by count (only for interval and ratio variables).
    • Median: Middle value in ordered data; if no middle exists, average the two middle values.
    • Mode: Most frequently occurring value.
    • Quartiles:
    • Q1: First quartile (25% of data).
    • Q2: Median (50% of data).
    • Q3: Third quartile (75% of data).

  • Measures of Spread:

    • Variance: Average of the squared differences from the mean.
    • Standard Deviation: Square root of variance, indicating average deviation from the mean.
    • Range: Difference between maximum and minimum scores.

Box Plots

  • Components:
    • Minimum
    • First Quartile (Q1)
    • Median (Q2)
    • Third Quartile (Q3)
    • Maximum

Probability Density Functions and Normal Distributions

  • Properties of Density Curves:

    • Describes the pattern of a quantitative variable.
    • Area under the curve equals 1 (representing total probability).

  • Normal Distribution Characteristics:

    • Symmetrical shape
    • Single peak (mean)
    • Bell-shaped curve

  • Notation: Denoted as N(µ, σ) where µ is mean and σ is standard deviation.

  • Relevance: Often applicable to real-world data, providing insights on probabilities and statistical conclusions.